The invention relates to the processing of signals for the exact measurement of a propagation time difference. The invention also relates to flow measuring apparatus, in particular to flow measuring apparatus for detecting the airflow during inspiration and expiration, which utilize the capture effect of ultrasound. Such flow measuring apparatus are also referred to as spirometers.
A number of apparatus are known, which utilize the capture effect of sound waves in fluids, especially air. There are substantially two construction types, which are illustrated in FIGS. 11 and 12.
The flow measuring apparatus 1 illustrated in FIG. 11 comprises a tube 2 as well as a first ultrasonic transducer 4 and a second ultrasonic transducer 5. The tube has a diameter d and an axis 3. Both ultrasonic transducers 4 and 5 are disposed on a second axis 6 so that each of the two transducers picks up the ultrasound generated by the respective other transducer. The second axis 6 encloses with axis 3 of the tube an angle α. If the medium in tube 2 is at rest, the sound propagation times t45 and t54 from ultrasonic transducer 5 to ultrasonic transducer 5 and from ultrasonic transducer 5 to ultrasonic transducer 4, respectively, are defined by
                              t          45                =                              t            54                    =                      L            c                                              (        1        )            with L designating the distance between both ultrasonic transducers and c the sound velocity in the medium at rest.
A flow in the tube 2 toward the right shortens the sound propagation time t45 from ultrasonic transducer 4 to ultrasonic transducer 5. At the same time, the sound propagation time t54 from ultrasonic transducer 5 to ultrasonic transducer 4 is extended. A rectangular flow profile is defined by
                              t          45                =                  L                      c            +                          v              ·                              cos                ⁡                                  (                  α                  )                                                                                        (        2        )                                          t          54                =                  L                      c            -                          v              ·                              cos                ⁡                                  (                  α                  )                                                                                        (        3        )            with v designating the velocity of the fluid toward the right. By introducing the propagation time difference Δt and the mean propagation time t0
                              Δ          ⁢                                          ⁢          t                ≡                              t            54                    -                      t            45                                              (        4        )                                          t          0                ≡                                            t              54                        +                          t              45                                2                                    (        5        )            one obtains for the flow velocity v:
                    v        =                              c                          2              ⁢                                                          ⁢                              cos                ⁡                                  (                  α                  )                                                              ⁢                                    Δ              ⁢                                                          ⁢              t                                      t              0                                                          (        6        )            
The sound velocity in the medium at rest can be calculated from (2) and (3):
                    c        =                  L                                    t              0                        -                                          1                4                            ⁢                                                Δ                  ⁢                                                                          ⁢                                      t                    2                                                                    t                  0                                                                                        (        7        )            
By inserting formula (7) into (6) one obtains:
                              V          .                =                              C            1                    ⁢                      L                          2              ⁢                                                          ⁢                              cos                ⁡                                  (                  α                  )                                                              ⁢                                    Δ              ⁢                                                          ⁢              t                                                      t                0                2                            -                                                1                  4                                ⁢                Δ                ⁢                                                                  ⁢                                  t                  2                                                                                        (        8        )            
The constant C1 is here the conversion factor between the flow velocity v and the flow rate {dot over (V)}. The constant C1 substantially includes the cross-sectional surface of tube 2, wherein r designates the radius of tube 2.C1=C2πr2  (9)
The constant C2 stands for a rectangular flow profile 1. In reality a velocity profile is formed in the tube, so that the velocity of the fluid depends on the position within the tube, on the flow rate and on the viscosity of the fluid. With respect to typical flow rates, the medium air and a tube diameter of two to three centimeters a laminar or turbulent flow range is formed at the wall of the tube and a turbulent flow range in the center of the tube. The greater the flow rate, the smaller is the laminar range and the greater is the turbulent range. This flow profile is responsible for non-linearities in the characteristic of the sensor.
The flow in the tube can be calculated from the propagation times t45 and t54 or from the propagation time difference Δt and the mean propagation time t0. This measuring principle is described, inter alia, in EP 0 051 293 A1, EP 0 243 515 A1, EP 0 597 060 B1, CH 669 463 A5, WO 00/26618 A1 and EP 0 713 080 A1. The smaller the angle α, the greater is the propagation time difference at the same flow.
The prior flow measuring apparatus 11 illustrated in FIG. 12 includes a U-shaped tube, which is comprised of a horizontal leg 12 and two vertical legs 17 and 18. The flow measuring apparatus 11 moreover includes ultrasonic transducers 14 and 15 disposed on an axis 13. The axis 13 is, at the same time, the axis of the horizontal leg 12. This structure has the advantage that the flow direction extends parallel to the axis of both ultrasonic transducers, thereby turning the angle α to 0° and, thus, to cos(α)=1. A disadvantage is the complicated construction as well as eddies which occur in the transition range between the vertical and the horizontal legs. Such a type of construction is described, for example, in WO 90/05283 A1 or EP 1 279 368 A2.
Irrespective of the constructional types it is known (e.g. WO 90/05283, EP 1 279 368 A2) to provide the ultrasonic transducers with acoustic impedance converters so as to improve the sound transmission between the fluid and the ultrasonic transducer and vice versa.
The state and the composition of the gas to be measured varies during the respiration. The ambient conditions are referred to as Ambient Temperature Pressure (ATP), which can be converted to Body Temperature Pressure Saturated (BTPS). Exhaled gas has a temperature of about 34° C. to 37° C. and is completely saturated with water vapor, that is, it corresponds to BTPS.
Additionally known is the sing-around technique, for example, from EP 0 713 080 A1. In the sing-around technique sound pulses are repeatedly transmitted from one ultrasonic transducer to the other ultrasonic transducer. As soon as a pulse is received, the next pulse is transmitted. After 1 to 200 pulses the direction is reversed. In this method, the resolution per time unit of the sound propagation time is increased at the expense of the measuring rate. This is particularly significant if media with a high sound velocity are concerned, in particular liquids, or if the propagation time can only be determined by involving, for technical reasons, intensive noise.
To analyze especially functions or signals that vary with time, for example, the correlation analysis, the Fourier analysis or transformation, the short-time Fourier transformation and the wavelet transformation are known.
The correlation coefficient φ of two functions f(t) and g(t) depending on time t and being defined at an interval of T1 to T2, is defined as:
                    φ        =                                            ∫                              T                1                                            T                2                                      ⁢                                          f                ⁡                                  (                  t                  )                                            ⁢                              g                ⁡                                  (                  t                  )                                            ⁢                              ⅆ                t                                                                                        ∫                                  T                  1                                                  T                  2                                            ⁢                                                                    f                    2                                    ⁡                                      (                    t                    )                                                  ⁢                                  ⅆ                  t                                ⁢                                                      ∫                                          T                      1                                                              T                      2                                                        ⁢                                                                                    g                        2                                            ⁡                                              (                        t                        )                                                              ⁢                                          ⅆ                      t                                                                                                                              (        10        )            
The correlation coefficient may be interpreted as a quantity for the similarity of two functions.
Moreover, the cross-correlation function φ (KKF) is defined as time average of the internal product of the two functions f(t) and g(t) shifted against each other by the time T:
                              Φ          ⁡                      (            τ            )                          =                              1                                          T                2                            -                              T                1                                              ⁢                                    ∫                              T                1                                            T                2                                      ⁢                                          f                ⁡                                  (                  t                  )                                            ⁢                              g                ⁡                                  (                                      t                    +                    τ                                    )                                            ⁢                              ⅆ                t                                                                        (        11        )            
By Fourier transformation, periodical functions with the period (T2-T1) can entirely be represented as weighted sum of sine and cosine functions. The Fourier coefficients can be comprehended as correlation factor of a periodic function and the base vectors of the Fourier space.
The Fourier transformation is predestinated to provide global information about a signal f(t) because its base functions extend over the entire time domain and do not allow any temporal locating ability.
To analyze transient or time-limited signals, the short-time Fourier transformation (STFT) was developed, which corresponds to a development from a one-dimensional time function into the two-dimensional time-frequency level. The signal is here divided into ranges which are assumed to be quasi-stationary, which are subjected to a windowed Fourier transformation independently of each other (Jürgen Niederholz, Anwendungen der Wavelet-Transformation in Übertragungssystemen, University of Duisburg, 1999 (http://www.ub.uni-duisburg.de/diss/diss0016/inhalt.htm). The window function in the time domain likewise causes a windowing in the frequency domain so that the STFT has a constant effective bandwidth Δω over the total frequency domain and an effective temporal expansion Δt over the total time domain.
The constant effective bandwidth and the restriction to sine and cosine functions were still perceived as a limitation under the STFT. Both restrictions are removed under the wavelet transformation. A general introduction into the wavelet transformation is given by Antje Ohihoff, Anwendung der Wavelettransformation in der Signalverarbeitung, University of Bremen, 1996. The function ψ(t) represents a valid wavelet if a real, finite constant Cψ exists, wherein {tilde over (Ψ)} (ω) is the Fourier transform of functions ψ(t).
                              C          Ψ                =                              ∫                          -              ∞                        ∞                    ⁢                                                                                                                                  Ψ                      ~                                        ⁡                                          (                      ω                      )                                                                                        2                            ω                        ⁢                          ⅆ              ω                                                          (        12        )            
One essential idea of the wavelet transformation consists in skillfully choosing the function ψ(t) and in adapting it to the problem to be solved, so that the functions to be analyzed can be described with a small number of wavelets. In other words, only a small number of coefficients of the base functions is to considerably differ from 0.
Based on a base wavelet or mother wavelet the remaining base functions, which form a complete basis for the function space, are generated with a translation parameter b and a dilatation parameter a according to equation (13).
                                                                        Ψ                                  a                  ,                  b                                            ⁡                              (                t                )                                      =                                          1                                                                          a                                                                                  ⁢                              Ψ                ⁡                                  (                                                            t                      -                      b                                        a                                    )                                                              ;          a                ,                  b          ∈          ℛ                ,                  a          ≠          0                                    (        13        )            
By equation (13) the so-called quality Q=ωM/Δω over the frequency domain is kept constant which, in the context of wavelet transformation, is also called scaling range. ωM is here a center frequency. In nature many phenomena occur, the quality of which is constant.